교과서 질문
In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. csc (cot⁻¹ x)
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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
모든 교과서
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 88
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 882장, 문제 88
In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sec (cos⁻¹ 1/x)
검증된 단계별 안내1
Recognize that the expression is \( \sec(\cos^{-1}(1/x)) \). Let \( \theta = \cos^{-1}(1/x) \), which means \( \cos(\theta) = \frac{1}{x} \).
Since \( \theta \) is an angle in a right triangle, draw a right triangle where the adjacent side to \( \theta \) is 1 and the hypotenuse is \( x \) (because \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)).
Use the Pythagorean theorem to find the length of the opposite side: \( \text{opposite} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).
Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}} \). Using the triangle, \( \sec(\theta) = \frac{x}{1} = x \).
Therefore, \( \sec(\cos^{-1}(1/x)) \) can be expressed algebraically as \( x \).
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Inverse Trigonometric Functions
Inverse trigonometric functions, like cos⁻¹ (arccos), return an angle whose trigonometric ratio matches the given value. For cos⁻¹(1/x), the output is an angle θ such that cos(θ) = 1/x, with θ typically in the range [0, π]. Understanding this helps translate the expression into a geometric context.
추천 영상:
4:28
Introduction to Inverse Trig Functions
Right Triangle Definitions of Trigonometric Ratios
Trigonometric ratios such as secant are defined using right triangles: sec(θ) = hypotenuse/adjacent. By interpreting θ = cos⁻¹(1/x), we can construct a right triangle where the adjacent side is 1 and the hypotenuse is x, allowing us to express sec(θ) algebraically in terms of x.
추천 영상:
5:19Solving Right Triangles with the Pythagorean Theorem
Algebraic Manipulation Using the Pythagorean Theorem
To find the missing side of the triangle, use the Pythagorean theorem: hypotenuse² = adjacent² + opposite². Given hypotenuse = x and adjacent = 1, the opposite side is √(x² - 1). This enables expressing sec(θ) = hypotenuse/adjacent = x/1 = x, or other related expressions, purely algebraically.
추천 영상:
5:19Solving Right Triangles with the Pythagorean Theorem
관련 실천
교과서 질문
The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = sin⁻¹ x + π/2
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교과서 질문
In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sin (tan⁻¹ x)
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교과서 질문
In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. ___ sec (sin⁻¹ x/√x²+4)
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교과서 질문
In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. cos (sin⁻¹ 1/x)
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교과서 질문
In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. tan (cos⁻¹ x)
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